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Lecture
12
2023
.
0922
$4
.
9
Linear
Approximations
$4
.
10
Taylor
Polynomails
The
idea
of
linear
approximation
is
very
simple
:
For
a
smooth
function
fix)
near
x
=
a
,
we
use
its
tan ge nt
line
to
approximate
it
.
xY
Q
fla
+
ox)
2
yoy
*
3
wy-dy
=
Ex)
((X)
=
f(a)
+
f((a)(X
-
a)
↳Y
·
-
dy
=
fix)dx
P
fal
·
↓
s
x
=
dx
X
-
7
a
a
+
0x
How
to
write
the
tangent
line
quation
?
·
Passing
through
p
=
(a
,
f(al
·
with
slope
f'(a)
=>
Y-
fla)
=
f'(a)(X-
a)
we
redefine
such
by
Lixs
.
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Why
is
linear
approximation
important
?
·
fixs
might
be
very
non-linear/complicated
but
Lex)
gives
essential
behaviour
of
fixs
near
x
=
a
.
efficient
for
computation
If
it
is
not
good
enough
,
we
need
higher
order
approximation
i
Taylor
polynomail
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-
Example
Use
the
linearization
for
x
at
x
=
25
to
find
an
approximation
value
of
EC
.
d
fix
=
r
fix)
=
xs
=
The
tange nt
line
of
fixs
at
x
=
25
is
the
line
passing
though
P
=
(25
,
55)
with
slope
=
=
(25)
-=
to
So
(Ix
=
55 +
to(X-25)
.
Putting
x
=
26
50
=
f(26)
=
((26)
=
55
+
%(26
-
25)
=
5
.
1
.
1]
Remark
The
error
term
is
defined
by
Error
&
true
value
-
approximation
=(x) If
f(x)
-
L(x)
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Theorem
11
<Error
formula
for
linearization)
suppose
that
f"its
exi st s
for
an
inter val
I
and
a
,
x
=
I
.
Then
Is
between
a
and
x
such
that
E(x1
=
I
(x-
as
,
with
EIx
fix-LIX)
,
and
2x1
=
f(a)
+
fi(a)(x-
a)
.
Proof
:
we
can
assume
x
>
a
.
The
care
x<a
is
similar
.
Then
apply
the
Generalized
Mean-value
Theorem
E(X)
EIX)
-
E(a)
=(U)
=--fins-fial
--
-
-
(x-
a)
X-as-
(a
-
as
214-a)
214
-
a)
↑
for
some
u
=
(a
,
x)
=
I
-"(s)
and
s
=
(a
.
4)
I
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Now
f(x)
=
LIx)
+
EIX)
=
LIx)
+
I
(x -
as
So
If
f"(t)
>0
fort
beteen
a
and
X
Convex
S
then
fix)
>
Lix)
-
If
f"(t)
<
0
fort
beteen
a
and
x
then
fix)
<
L(X)
concave
-
Theorem
If
f(x)
<
0
for
all
xe
I
interval
then
o
is
conv e x
on
1.
~
If
f) <0
for
all
x
=
1
,
then
f
is
concav e
.
-
Recall
:
I
is
called
convex/concave
on
I
if
F
X
,,
x2
=
2
,
and
0 ft=1
,
we
have
f(tx
,
+
x
-
t)x 2)
=
(
,
)
tf(xx)
+
(1
-
t)f(x2)
.
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Example
·
Let
fix)
=
ex
,
g(x)
=
en(X)
,
hix1
=
x
!
Study
their
convex/concave
proper ty
.
·
How
about
wixi=
a
Y
for
some
positive
constant
a
?
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S
Taylor
Polynomails
The
idea
is
also
simple
:
For
a
smooth
function
y
=
fix)
near
x
=
a
·
The
dominate
approximation
is
given
by
linear
term
of
(x-as
more
precisely
flas+
fas(x-as(
·
If
we
use
higher
order
term
(x-as"
we
could
get
a
better
approximation
·
what
is
the
coefficient
before
(x-as"
?
(a)
2
·
Even
higher
order
term
?
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Definition
If
f(x)
exists
on
an
open
inter val
containing
x
=
a
,
then
the
polynomail
Pnexs
+
fial
+
1
"
(x -
a)
+
(x-
as
+
...
+
"(x-
is
called
the
n-th
order
Taylor
polynomal
for
-
near
a.
Prop erty
.
Puixs
matches
fix,
at
a
up
to
order
n
.
More
precisely
:
Pr(a)
=
f(a)
Pn(a)
=
-(a)
pas
fas
.
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And
we
have
f(x)
=
PnIX)
.
Define
En1x)
=
fix)
-
PuIX)
n-th
order
Error
term
Theorem
12
If
the
(n+1)
-
th
order
derivative
fint)
,
-)
exists
for
+
in
an
inter val
containing
x
and
a
,
then
(((x
-
ag+
En(x1
=
(n+1)
!
for
come
between
a
and
X
.
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Def
(Big
-O
Notation)
We
write
f(x)
=
0
(ux)
as
x
-
a
if
/fixs/a
Kluvsl
holds
for
X
in
a
neibourhood
of
a
and
some
constant
12
.
Example
.
As
x-O
3
ae
*
=
1
+
x
+
!
+
+
..
+
+
0x)
2
X
xY
(b)
20SX
=
1
-
I
!
I
!..
+
E1)"
+
Olx2nt'
I
x
5
1
sinx
=
x
-
+
5
!
+
...
+
7)
,
+01x
id
x
=
1
+
x
+
x4
...
+
x"
+
0(x4+)
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(e)
x
=
I
=
1
+
(
-
x)
+
1
-
x4
-
..
+
(
-
x
+
0(x+)
ie
.
x
= 1
-
x
+
x
+
..
.
+
ex
+
0xnt
,
If)
InC1x
=
x
-
+
+
..
+
74
*
+
0x4)
x5
191
tan"(x)
=
x
-
=
+
5
-
...
+
+
0x4
(-1)
(h)
(1
+
x)
*
=
H
xx
+
"x4
...
+
a
-
+
x
+
0x
Taylor
series
Pu(X)
n
+
o
In
stead
of
order
a
polynomial
,
we
are
using
a
series
·
Theorem
(Uniqueness)
Assume
the
exis tance
of
Taylor
polynomial
(or
Taylor
series]
near
x
=
a
then
Taylor
polynomial
(or
Taylor
Series)
is
unique
.
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Example
Evaluate
2
Sinx-sin(2x)
lim
29
x
-
2
-
2x
-
x "
X
+
0
A
ation
2
.
(x
-
+
01x)
-
12x)
-
2
+
0153)
lim
X
+
O
211
+
x
+
+
+
01x4))
-
2 -
2x
-
x
x
+
54x
+
0(x)
-
3
=lim
x
-
0
x
+
0(xY)
5
=
(5
+
2)
+
0IxY
-
*
=
3
.
-
lim
I
5
+
O(X)
5
x
+
0
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Example
Find
the
Taylor
polynomial
of
tanh"(x)
near
x-o
up
to
order
2n+1
.
Hint
tanh"(x)=
Ien
(
*
S
1
tanh"
(x)
=
I
In
1
*
(
=>
I((n[1
+
x)
-
(((
-
x))
x2
=
I
I
X
-
I
+
...
+
(1)2x4
+
01x2
2n+1
,
+xx
2n
-
(x
-
I
.....
-
)
***
+
01x
+
]
even
terms
vanish
I
0
+
2
.
X
2n
+
1
=
I
[2
.
x
+
0
+
2
..
+
-
+
0(x2+]
4
2n+1
odd
terms
survive
3
=
x
+
+
+
...
+
+
0xxn
+ ,
1
